Bridging the micro-Hz gravitational wave gap via Doppler tracking with the Uranus Orbiter and Probe Mission: Massive black hole binaries, early universe signals and ultra-light dark matter

With the recent announcement by NASA's \textit{Planetary Science and Astrobiology Decadal Survey 2023-2032}, a priority flagship mission to the planet Uranus is anticipated. Here, we explore the prospects of using the mission's radio Doppler tracking equipment to detect gravitational waves (GWs) and other analogous signals related to dark matter (DM) over the duration of its interplanetary cruise. We develop a methodology to stack tracking data and account for time varying detector geometry, thereby constructing the sensitivity of the mission to GWs over the wide frequency range of $3\times 10^{-9}$ Hz to $10^{-4}$ Hz. We find that the mission has the potential to fill the gap between pulsar timing and space-based-interferometry GW observatories. If improvements in reducing \textit{Cassini} era noise by a factor of $\sim$10 are implemented, we forecast the detection of $\mathcal{\mathcal{O}}(\rm{few})$ individual massive black hole binaries using two independent population models. Additionally, we determine the mission's sensitivity to both astrophysical and primordial stochastic gravitational wave backgrounds, as well as its capacity to test, or even confirm via detection, ultralight DM models. In all these cases, the tracking of the spacecraft over its interplanetary cruise would enable coverage of unexplored regions of parameter space, where signals from new phenomena in our Universe may be lurking.